PRMO 2016 Problem 1 | Number Theory Problem

Consider all possible integers $n \geq 0$ such that
$$
\left(5 \times 3^{m}\right)+4=n^{2}
$$
holds for some corresponding integer $m \geq 0$. Find the sum of all such $n$.

Elementary Number Theory by David Burton

Excursion in mathematics

PRMO 2016

The required answer is 10

$
\left(5 \times 3^{m}\right)+4=n^{2}
$

$\rightarrow \left(5 \times 3^{m}\right) = n^{2}-4$

$\rightarrow \left(5 \times 3^{m}\right) = (n-2)(n+2)$

Observe that $gcd (n-2,n+2)$ = $1$ or $2$ or $4 $

But $\left(5 \times 3^{m}\right) = (n-2)(n+2)$

Hence both $(n-2)$ , $(n+2)$ are odd.

Hence one possible case:

$(n-2)$ = $5$

$(n+2)$ = $3^{m}$

$\rightarrow$ $4$ = $3^{m}$ - $5$

$\rightarrow$ $m$ = $2$

$\rightarrow$ $n$ = $7$

Similarly find other values of n and add

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